Taylor and Maclaurin Series

Taylor series

About x = a

f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} (x - a)^{n}

Lagrange error bound

| R_{n} (x) | \leq \frac{max | f^{(n + 1)} (z) |}{(n + 1)!} | x - a |^{n + 1}

Geometric

\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots

|x| < 1

e to the x

e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots

Sine

\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots

Cosine

\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots

Natural log

\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots

−1 < x ≤ 1

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